II.1 Age-Structured and Stage-Structured Population Dynamics
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II.1 Age-Structured and Stage-Structured Population Dynamics Mark Rees and Stephen P. Ellner OUTLINE 1. Age-structured models: Life tables and the Leslie matrix 2. Stage-structured matrix models 3. Integral projection models 4. Continuous-time models with age structure 5. Applications and extensions 6. Coda When all individuals in a population are identical, we can characterize the population just by counting the number of individuals. However, the individuals within many animal and plant populations differ in important ways that inﬂuence their current and future prospects of survival and reproduction. For example, larger individuals typically have greater chances of survival, produce more and sometimes larger offspring, and often have slower growth rates. In such cases, characterizing the population structure—the numbers of individuals of each different type—is critical for understanding how the population will change through time. In this chapter, we examine some of the main types of models used for describing and forecasting the dynamics of structured populations. Age-structured models in discrete time, appropriate for populations in seasonal environments, were developed centuries ago by the great mathematician Leonhard Euler (1707–1783). These are considered ﬁrst, before moving to models where individuals are characterized by their stage in the life cycle (e.g., seed versus ﬂowering plant, larva versus adult). Next we look at how to incorporate differences among individuals that vary continuously, such as size. Having explored discrete-time models, we brieﬂy turn to continuoustime models and then present some applications and extensions. GLOSSARY age structure. Distribution of ages in a population matrix. A rectangular array of symbols, which could represent numbers, variables, or functions 1. AGE-STRUCTURED MODELS: LIFE TABLES AND THE LESLIE MATRIX The simplest age-structured models assume that each individual’s chance of survival and reproduction depends only on its age; there are no effects of population density. The standard model counts only females (assuming no shortage of mates) and assumes that all births occur in a single birth pulse immediately before the population is censused (a so-called postbreeding census). The population dynamics is then summarized by the following equations: n0(t þ 1) ¼ f0n0(t) þ f1n1(t) þ f2n2(t) þ   ¼ X A a ¼ 0 fana(t) na(t þ 1) ¼ pa  1na  1(t), (1) where na(t) is the number of individuals of age a at time t, fa, and pa are, respectively, the average fecundity and the probability of survival to age a þ 1 of age a individuals , and A is the maximum possible age (or the maximum age at which reproduction occurs, if postreproductives are omitted from the population count). Because births occur just before the next census, fa ¼ pama þ 1, where ma þ 1 is the number of offspring produced by an age a þ 1 female. Another way of formulating the model is to assume a prebreeding census, so the population is censused immediately before the birth pulse. This has two important consequences: (1) all individuals are at least age 1, and (2) in this case fa ¼ p0ma, so fecundity depends on the number of offspring produced now, ma, and the chance that they survive to be censused at age 1, p0. The simple age-structured model can be written as a matrix, commonly known as a Leslie matrix after British ecologist P. H. Leslie. Expressing equation 1 in matrix form simply means putting the fs and ps in the right places: n0(t þ 1) n1(t þ 1) . . . nA(t þ 1) 2 6 6 6 6 6 4 3 7 7 7 7 7 5¼ f0 f1 f2    fA p0 0 0    0 0 p1 0    0 . . . . . . .. . . . . 0 0 pA  1 0 2 6 6 6 6 6 4 3 7 7 7 7 7 5 n0(t) n1(t) . . . nA(t) 2 6 6 6 6 6 4 3 7 7 7 7 7 5 (2) or, more compactly, n(t þ 1) ¼ Ln(t), (3) where L is the matrix is equation 2. When L is a matrix with n columns and n(t) a column vector of length n, then Ln(t) is a vector whose ith element is [Ln(t)]i ¼ X n j ¼ 1 Lijn(t)j, (4) where Lij is the number in the ith row and jth column of L. Matrix multiplication expresses equation 1 as a single operation; it also means that the tools of linear algebra can be used to study how the population varies through time. Now that we have formulated the model, how does it behave? To...